Question: Find the least integer value of $x$ for which $2|x| + 7 < 17$.
Solution: First, solve the inequality so that only the absolute value quantity is on the left and the constant value is on the right.

\begin{align*}
2|x| + 7&< 17\\
2|x|&<10\\
|x|&<5
\end{align*}To solve the inequality which has an absolute value in it, we must turn this into two different inequalities, one as normal, one with a reversed sign and opposite resulting value.  Both will have the absolute value removed.

\begin{align*}
x &< 5 \\
x &> -5
\end{align*}Since we need the least integer value of $x$, and $x$ has to be $\textbf{greater than }$ -5, the next smallest integer is $\boxed{-4}$.